Rendering does not process transparency changes over time - Playing my timeline do. v2.79b
Thanks for the file, i didnt realise that you were using Blender Render, i cant help you with that because i only use cycles and eevee.  

If you want the retro-aesthetics of blender you could even use eevee and configure it that way, plus benfit from the rendering speed.  

In Cycles as in EEVEE you would do your animation via the Node-Editor and animate the facture of a mix shader that hast one diffuse and one transparent shader (with absolute white!) in it.